E04JCF (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

E04JCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E04JCF is an easy-to-use algorithm that uses methods of quadratic approximation to find a minimum of an objective function

F

over

x \in R^{n}

, subject to fixed lower and upper bounds on the independent variables

x_{1}, x_{2}, \dots, x_{n}

. Derivatives of

F

are not required.

The routine is intended for functions that are continuous and that have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities). Efficiency is maintained for large

n

.

2 Specification

SUBROUTINE E04JCF (

OBJFUN, N, NPT, X, BL, BU, RHOBEG, RHOEND, MONFUN, MAXCAL, F, NF, IUSER, RUSER, IFAIL)

INTEGER	N, NPT, MAXCAL, NF, IUSER(*), IFAIL
REAL (KIND=nag_wp)	X(N), BL(N), BU(N), RHOBEG, RHOEND, F, RUSER(*)
EXTERNAL	OBJFUN, MONFUN

3 Description

E04JCF is applicable to problems of the form:

\underset{x \in R^{n}}{minimize} F (x) subject to ℓ \leq x \leq u and ℓ \leq u,

where

F

is a nonlinear scalar function whose derivatives may be unavailable, and where the bound vectors are elements of

R^{n}

. Relational operators between vectors are interpreted elementwise.

Fixing variables (that is, setting

ℓ_{i} = u_{i}

for some

i

) is allowed in E04JCF.

You must supply a subroutine to calculate the value of

F

at any given point

x

.

The method used by E04JCF is based on BOBYQA, the method of Bound Optimization BY Quadratic Approximation described in Powell (2009). In particular, each iteration of E04JCF generates a quadratic approximation

Q

to

F

that agrees with

F

at

m

automatically chosen interpolation points. The value of

m

is a constant prescribed by you. Updates to the independent variables mostly occur from approximate solutions to trust-region subproblems, using the current quadratic model.

4 References

Powell M J D (2009) The BOBYQA algorithm for bound constrained optimization without derivatives Report DAMTP 2009/NA06 University of Cambridge http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf

5 Parameters

1: OBJFUN – SUBROUTINE, supplied by the user.External Procedure

OBJFUN must evaluate the objective function

F

at a specified vector

x

.

The specification of OBJFUN is:

SUBROUTINE OBJFUN (

N, X, F, IUSER, RUSER, INFORM)

INTEGER	N, IUSER(*), INFORM
REAL (KIND=nag_wp)	X(N), F, RUSER(*)

1: N – INTEGERInput: On entry: $n$ , the number of independent variables.
2: X(N) – REAL (KIND=nag_wp) arrayInput: On entry: $x$ , the vector at which the objective function is to be evaluated.
3: F – REAL (KIND=nag_wp)Output: On exit: must be set to the value of the objective function at $x$ , unless you have specified termination of the current problem using INFORM.
4: IUSER( $*$ ) – INTEGER arrayUser Workspace
5: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: OBJFUN is called with the parameters IUSER and RUSER as supplied to E04JCF. You are free to use the arrays IUSER and RUSER to supply information to OBJFUN as an alternative to using COMMON global variables.
6: INFORM – INTEGEROutput: On exit: must be set to a value describing the action to be taken by the solver on return from OBJFUN. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.

OBJFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04JCF is called. Parameters denoted as Input must not be changed by this procedure.

2: N – INTEGERInput

On entry:

n

, the number of independent variables.

Constraint:

N \geq 2

and

n_{r} \geq 2

, where

n_{r}

denotes the number of non-fixed variables.

3: NPT – INTEGERInput

On entry:

m

, the number of interpolation conditions imposed on the quadratic approximation at each iteration.

Suggested value:

NPT = 2 \times n_{r} + 1

, where

n_{r}

denotes the number of non-fixed variables.

Constraint:

n_{r} + 2 \leq NPT \leq \frac{(n_{r} + 1) \times (n_{r} + 2)}{2}

, where

n_{r}

denotes the number of non-fixed variables.

4: X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an estimate of the position of the minimum. If any component is out-of-bounds it is replaced internally by the bound it violates.

On exit: the lowest point found during the calculations. Thus, if

IFAIL = 0

on exit, X is the position of the minimum.

5: BL(N) – REAL (KIND=nag_wp) arrayInput

6: BU(N) – REAL (KIND=nag_wp) arrayInput

On entry: the fixed vectors of bounds: the lower bounds

ℓ

and the upper bounds

u

, respectively. To signify that a variable is unbounded you should choose a large scalar

r

appropriate to your problem, then set the lower bound on that variable to

- r

and the upper bound to

r

. For well-scaled problems

r = r_{\max}^{\frac{1}{4}}

may be suitable, where

r_{\max}

denotes the largest positive model number (see X02ALF).

Constraints:

if $X (i)$ is to be fixed at $BL (i)$ , then $BL (i) = BU (i)$ ;
otherwise $BU (i) - BL (i) \geq 2.0 \times RHOBEG$ , for $i = 1, 2, \dots, N$ .

7: RHOBEG – REAL (KIND=nag_wp)Input

On entry: an initial lower bound on the value of the trust-region radius.

Suggested value: RHOBEG should be about one tenth of the greatest expected overall change to a variable: the initial quadratic model will be constructed by taking steps from the initial X of length RHOBEG along each coordinate direction.

Constraints:

$RHOBEG > 0.0$ ;
$RHOBEG \geq RHOEND$ .

8: RHOEND – REAL (KIND=nag_wp)Input

On entry: a final lower bound on the value of the trust-region radius.

Suggested value: RHOEND should indicate the absolute accuracy that is required in the final values of the variables.

Constraint:

RHOEND > 0.0

.

9: MONFUN – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

MONFUN may be used to monitor the optimization process. It is invoked every time a new trust-region radius is chosen.

If no monitoring is required, MONFUN may be the dummy monitoring routine E04JCP supplied by the NAG Library.

The specification of MONFUN is:

SUBROUTINE MONFUN (

N, NF, X, F, RHO, IUSER, RUSER, INFORM)

INTEGER	N, NF, IUSER(*), INFORM
REAL (KIND=nag_wp)	X(N), F, RHO, RUSER(*)

1: N – INTEGERInput: On entry: $n$ , the number of independent variables.
2: NF – INTEGERInput: On entry: the cumulative number of calls made to OBJFUN.
3: X(N) – REAL (KIND=nag_wp) arrayInput: On entry: the current best point.
4: F – REAL (KIND=nag_wp)Input: On entry: the value of OBJFUN at X.
5: RHO – REAL (KIND=nag_wp)Input: On entry: a lower bound on the current trust-region radius.
6: IUSER( $*$ ) – INTEGER arrayUser Workspace
7: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: MONFUN is called with the parameters IUSER and RUSER as supplied to E04JCF. You are free to use the arrays IUSER and RUSER to supply information to MONFUN as an alternative to using COMMON global variables.
8: INFORM – INTEGEROutput: On exit: must be set to a value describing the action to be taken by the solver on return from MONFUN. Specifically, if the value is negative the solution of the current problem will terminate immediately; otherwise, computations will continue.

MONFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04JCF is called. Parameters denoted as Input must not be changed by this procedure.

10: MAXCAL – INTEGERInput

On entry: the maximum permitted number of calls to OBJFUN.

Constraint:

MAXCAL \geq 1

.

11: F – REAL (KIND=nag_wp)Output

On exit: the function value at the lowest point found (X).

12: NF – INTEGEROutput

On exit: unless

IFAIL = 1

or

- 999

on exit, the total number of calls made to OBJFUN.

13: IUSER( $*$ ) – INTEGER arrayUser Workspace

14: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by E04JCF, but are passed directly to OBJFUN and MONFUN and may be used to pass information to these routines as an alternative to using COMMON global variables.

15: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

,

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

E04JCF returns with

IFAIL = 0

if the final trust-region radius has reached its lower bound RHOEND.

6 Error Indicators and Warnings

If on entry

IFAIL = 0

or

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: An input parameter is invalid.

$IFAIL = 2$: The function evaluations limit was reached: OBJFUN has been called MAXCAL times.

$IFAIL = 3$: The predicted reduction in a trust-region step was non-positive. Check your specification of OBJFUN and whether the function needs rescaling. Try a different initial X.

$IFAIL = 4$: A rescue procedure has been called in order to correct damage from rounding errors when computing an update to a quadratic approximation of $F$ , but no further progess could be made. Check your specification of OBJFUN and whether the function needs rescaling. Try a different initial X.

$IFAIL = 5$: You terminated the solver.

You indicated that you wished to halt solution of the current problem by setting INFORM to a negative value in either OBJFUN or MONFUN.

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

Experience shows that, in many cases, on successful termination the

\infty

-norm distance from the best point

x

to a local minimum of

F

is less than

10 \times RHOEND

, unless RHOEND is so small that such accuracy is unattainable.

8 Further Comments

For each invocation of E04JCF, local workspace arrays of fixed length are allocated internally. The total size of these arrays amounts to

(NPT + 6) \times (NPT + n_{r}) + \frac{n_{r} \times (3 n_{r} + 21)}{2}

real elements and

n_{r}

integer elements, where

n_{r}

denotes the number of non-fixed variables; that is, the total size is

O (n_{r}^{4})

. If you follow the recommendation for the choice of NPT on entry, this total size reduces to

O (n_{r}^{2})

.

Usually the total number of function evaluations (NF) is substantially less than

O (n_{r}^{2})

, and often, if

NPT = 2 \times n_{r} + 1

on entry, NF is only of magnitude

n_{r}

or less.

9 Example

This example involves the minimization of

F = {(x_{1} + 10 x_{2})}^{2} + 5 {(x_{3} - x_{4})}^{2} + {(x_{2} - 2 x_{3})}^{4} + 10 {(x_{1} - x_{4})}^{4}

subject to

\begin{matrix} 1 \leq x_{1} \leq 3 \\ - 2 \leq x_{2} \leq 0 \\ 1 \leq x_{4} \leq 3, \end{matrix}

starting from the initial guess

(3, - 1, 0, 1)

.

9.1 Program Text

Program Text (e04jcfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (e04jcfe.r)

E04JCF (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012